Spin-Orbit-Free Topological Insulators without Time-Reversal Symmetry

A. Alexandradinata, Chen Fang, Matthew J. Gilbert, and B. Andrei Bernevig

Phys. Rev. Lett. 113, 116403 – Published 12 September 2014

Abstract

We explore the 32 crystallographic point groups and identify topological phases of matter with robust surface modes. For $n = 3, 4$ , and 6 of the $C_{n v}$ groups, we find the first-known 3D topological insulators without spin-orbit coupling, and with surface modes that are protected only by point groups; i.e., the relevant symmetries are purely crystalline and do not include time reversal. To describe these $C_{n v}$ systems, we introduce the notions of (a) a halved mirror chirality, an integer invariant which characterizes half-mirror-planes in the 3D Brillouin zone, and (b) a bent Chern number, the traditional Thouless–Kohmoto–Nightingale–den Nijs invariant generalized to bent 2D manifolds. We find that a Weyl semimetallic phase intermediates two gapped phases with distinct halved chiralities. In addition to electronic systems without spin-orbit coupling, our findings also apply to intrinsically spinless systems such as photonic crystals and ultracold atoms.

Received 14 March 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.116403

Authors & Affiliations

A. Alexandradinata¹, Chen Fang^2,3,1, Matthew J. Gilbert^4,3, and B. Andrei Bernevig¹

¹Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
³Micro and Nanotechnology Laboratory, University of Illinois, 208 N. Wright Street, Urbana, Illinois 61801, USA
⁴Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA

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Vol. 113, Iss. 11 — 12 September 2014

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Images

Figure 1
Bottom: (a) Half-mirror-planes (HMPs) in the 3D Brillouin zone (BZ) of a hexagonal lattice with $C_{3 v}^{(b)}$ symmetry. Blue face that projects to $\bar{Γ} - \bar{K}$ : ${HMP}_{1}$ . Brown, $\bar{K} - {\bar{K}}^{'}$ : ${HMP}_{2}$ . Red, $\bar{Γ} - {\bar{K}}^{'}$ : ${HMP}_{3}$ . (b) HMPs in the 3D BZ of a tetragonal lattice with $C_{4 v}$ symmetry. Red face that projects to $\bar{Γ} - \bar{M}$ : ${HMP}_{4}$ . Blue, $\bar{Γ} - \bar{X} - \bar{M}$ : ${HMP}_{5}$ . (c) Blue face that projects to $\bar{Γ} - \bar{K}$ : ${HMP}_{1}$ in the 3D BZ of a hexagonal lattice with $C_{6 v}$ symmetry. Note that the black-colored submanifold in (c) is not a HMP. In each of (a),(b) and (c) we define a bent Chern number on the triangular pipe with its ends identified. Top: Nonblack lines are half-mirror lines (HMLs) in the corresponding 2D BZ of the 001 surface; each HML connects two distinct $C_{m}$ -invariant points with $m > 2$ .
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Figure 2
(a) Top-down view of hexagonal BZ with $C_{3 v}^{(a)}$ symmetry; our line of sight is parallel to the rotational axis. (b) Hexagonal BZ with $C_{3 v}^{(b)}$ symmetry. (c) Tetragonal BZ with $C_{4 v}$ symmetry. (d) Hexagonal BZ with $C_{6 v}$ symmetry. Reflection-invariant planes are indicated by solid lines. Except the line through $Γ$ , all nonequivalent $C_{n}$ -invariant lines are indicated by circles for $n = 2$ , triangles for $n = 3$ , and squares for $n = 4$ . For each of ${C_{3 v}^{(a)}, C_{3 v}^{(b)}}$ , there are two independent mirror Chern numbers, defined as $C_{e}$ ( $C_{o}$ ) in the mirror-even (odd) subspace [9]. In both (a) and (b), $C_{e}$ and $C_{o}$ are defined on a single mirror plane indicated in red; in (b), the two red lines correspond to two projected planes which connect through a reciprocal lattice vector (dashed arrow).
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Figure 3
Distinct connectivities of the (001) surface bands along the half-mirror lines. Black solid (dotted) lines indicate surface bands with eigenvalue $+ 1$ ( $- 1$ ) under reflection $M_{i}$ ; crossings between solid and dotted lines are robust due to reflection symmetry. For simplicity, we have depicted all degeneracies at momenta $s = 0$ and $s = 1$ as dispersing linearly with momentum. This is true if the little group of the wave vector (at $s \in {0, 1}$ ) is $C_{3 v}$ , but for $C_{4 v}$ and $C_{6 v}$ such crossings are in reality quadratic [9].
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Figure 4
(a)–(d) illustrate how the halved chirality of a HMP may change. The direction of arrows indicate whether Weyl nodes are created or annihilated. $+ (-)$ labels a Berry monopole with positive (negative) charge; $e$ ( $o$ ) labels a crossing in the HMP between mirror-even (-odd) bands. (e)–(g) In three examples, we provide a top-down view of the trajectories of Weyl nodes, in the transition between two distinct gapped phases. Our line of sight is parallel to $\hat{z}$ . Black lines indicate mirror faces, while colored lines specially indicate HMPs, with the same color legend as in Fig. 1. (e) describes a $C_{4 v}$ system, (f) $C_{3 v}^{(b)}$ , and (g) $C_{6 v}$ .
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